The main focus is now on writing up the results, with a draft paper close to completion here (with the directory of source files here). Most of the sections are now written up more or less completely, with the exception of the appendix on narrow admissible tuples, which was awaiting the bounds on such tuples to stabilise. There is now also an acknowledgments section (linking to the corresponding page on the wiki, which participants should check to see if their affiliations etc. are posted correctly), and in the final remarks section there is now also some discussion about potential improvements to the bounds. I’ve also added some mention of a recent paper of Banks, Freiberg and Maynard which makes use of some of our results (in particular, that ). On the other hand, the portions of the writeup relating to potential improvements to the MPZ estimates have been commented out, as it appears that one cannot easily obtain the exponential sum estimates required to make those go through. (Perhaps, if there are significant new developments, one could incorporate them into a putative Polymath8c project, although at present I think there’s not much urgency to start over once again.)

Regarding the numerics in Section 7 of the paper, one thing which is missing at present is some links to code in case future readers wish to verify the results; alternatively one could include such code and data into the arXiv submission.

It’s about time to discuss possible journals to submit the paper to. Ken Ono has invited us to submit to his new journal, “Research in the Mathematical Sciences“. Another option would be to submit to the same journal “Algebra & Number Theory” that is currently handling our Polymath8a paper (no news on the submission there, but it is a very long paper), although I think the papers are independent enough that it is not absolutely necessary to place them in the same journal. A third natural choice is “Mathematics of Computation“, though I should note that when the Polymath4 paper was submitted there, the editors required us to use our real names instead of the D.H.J. Polymath pseudonym as it would have messed up their metadata system otherwise. (But I can check with the editor there before submitting to see if there is some workaround now, perhaps their policies have changed.) At present I have no strong preferences regarding journal selection, and would welcome further thoughts and proposals. (It is perhaps best to avoid the journals that I am editor or associate editor of, namely Amer. J. Math, Forum of Mathematics, Analysis & PDE, and Dynamics and PDE, due to conflict of interest (and in the latter two cases, specialisation to a different area of mathematics)).

Numerical progress on these bounds have slowed in recent months, although we have very recently lowered the unconditional bound on from 252 to 246 (see the wiki page for more detailed results). While there may still be scope for further improvement (particularly with respect to bounds for with , which we have not focused on for a while, it looks like we have reached the point of diminishing returns, and it is time to turn to the task of writing up the results.

A draft version of the paper so far may be found here (with the directory of source files here). Currently, the introduction and the sieve-theoretic portions of the paper are written up, although the sieve-theoretic arguments are surprisingly lengthy, and some simplification (or other reorganisation) may well be possible. Other portions of the paper that have not yet been written up include the asymptotic analysis of for large k (leading in particular to results for m=2,3,4,5), and a description of the quadratic programming that is used to estimate for small and medium k. Also we will eventually need an appendix to summarise the material from Polymath8a that we would use to generate various narrow admissible tuples.

One issue here is that our current unconditional bounds on for m=2,3,4,5 rely on a distributional estimate on the primes which we believed to be true in Polymath8a, but never actually worked out (among other things, there was some delicate algebraic geometry issues concerning the vanishing of certain cohomology groups that was never resolved). This issue does not affect the m=1 calculations, which only use the Bombieri-Vinogradov theorem or else assume the generalised Elliott-Halberstam conjecture. As such, we will have to rework the computations for these , given that the task of trying to attain the conjectured distributional estimate on the primes would be a significant amount of work that is rather disjoint from the rest of the Polymath8b writeup. One could simply dust off the old maple code for this (e.g. one could tweak the code here, with the constraint 1080*varpi/13+ 330*delta/13<1 being replaced by 600*varpi/7+180*delta/7<1), but there is also a chance that our asymptotic bounds for (currently given in messy detail here) could be sharpened. I plan to look at this issue fairly soon.

Also, there are a number of smaller observations (e.g. the parity problem barrier that prevents us from ever getting a better bound on than 6) that should also go into the paper at some point; the current outline of the paper as given in the draft is not necessarily comprehensive.

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page.

The focus is now on bounding unconditionally (in particular, without resorting to the Elliott-Halberstam conjecture or its generalisations). We can bound whenever one can find a symmetric square-integrable function supported on the simplex such that

Our strategy for establishing this has been to restrict to be a linear combination of symmetrised monomials (restricted of course to ), where the degree is small; actually, it seems convenient to work with the slightly different basis where the are restricted to be even. The criterion (1) then becomes a large quadratic program with explicit but complicated rational coefficients. This approach has lowered down to , which led to the bound .

Actually, we know that the more general criterion

will suffice, whenever and is supported now on and obeys the vanishing marginal condition whenever . The latter is in particular obeyed when is supported on . A modification of the preceding strategy has lowered slightly to , giving the bound which is currently our best record.

However, the quadratic programs here have become extremely large and slow to run, and more efficient algorithms (or possibly more computer power) may be required to advance further.

This is the eighth thread for the Polymath8b project to obtain new bounds for the quantity

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page.

The big news since the last thread is that we have managed to obtain the (sieve-theoretically) optimal bound of assuming the generalised Elliott-Halberstam conjecture (GEH), which pretty much closes off that part of the story. Unconditionally, our bound on is still . This bound was obtained using the “vanilla” Maynard sieve, in which the cutoff was supported in the original simplex , and only Bombieri-Vinogradov was used. In principle, we can enlarge the sieve support a little bit further now; for instance, we can enlarge to , but then have to shrink the J integrals to , provided that the marginals vanish for . However, we do not yet know how to numerically work with these expanded problems.

Given the substantial progress made so far, it looks like we are close to the point where we should declare victory and write up the results (though we should take one last look to see if there is any room to improve the bounds). There is actually a fair bit to write up:

Improvements to the Maynard sieve (pushing beyond the simplex, the epsilon trick, and pushing beyond the cube);

Asymptotic bounds for and hence ;

Explicit bounds for (using the Polymath8a results)

;

on GEH (and parity obstructions to any further improvement).

I will try to create a skeleton outline of such a paper in the Polymath8 Dropbox folder soon. It shouldn’t be nearly as big as the Polymath8a paper, but it will still be quite sizeable.

This is the seventh thread for the Polymath8b project to obtain new bounds for the quantity

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page.

The current focus is on improving the upper bound on under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from to . Very recently, we have been able to exploit GEH more fully, leading to a promising new expansion of the sieve support region. The problem now reduces to the following:

Problem 1 Does there exist a (not necessarily convex) polytope with quantities , and a non-trivial square-integrable function supported on such that

when ;

when ;

when ;

and such that we have the inequality

An affirmative answer to this question will imply on GEH. We are “within two percent” of this claim; we cannot quite reach yet, but have got as far as . However, we have not yet fully optimised in the above problem. In particular, the simplex

is now available, and should lead to some noticeable improvement in the numerology.

There is also a very slim chance that the twin prime conjecture is now provable on GEH. It would require an affirmative solution to the following problem:

Problem 2 Does there exist a (not necessarily convex) polytope with quantities , and a non-trivial square-integrable function supported on such that

when ;

when ;

and such that we have the inequality

We suspect that the answer to this question is negative, but have not formally ruled it out yet.

For the rest of this post, I will justify why positive answers to these sorts of variational problems are sufficient to get bounds on (or more generally ).

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page (which has recently returned to full functionality, after a partial outage).

The current focus is on improving the upper bound on under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from to , which looks to be the limit of the method (see this previous comment for a semi-rigorous reason as to why is not possible with this method). With the most general Selberg sieve available, the problem reduces to the following three-dimensional variational one:

Problem 1 Does there exist a (not necessarily convex) polytope with quantities , and a non-trivial square-integrable function supported on such that

when ;

when ;

when ;

and such that we have the inequality

(Initially it was assumed that was convex, but we have now realised that this is not necessary.)

An affirmative answer to this question will imply on GEH. We are “within almost two percent” of this claim; we cannot quite reach yet, but have got as far as . However, we have not yet fully optimised in the above problem.

The most promising route so far is to take the symmetric polytope

with symmetric as well, and (we suspect that the optimal will be roughly ). (However, it is certainly worth also taking a look at easier model problems, such as the polytope , which has no vanishing marginal conditions to contend with; more recently we have been looking at the non-convex polytope .) Some further details of this particular case are given below the fold.

There should still be some progress to be made in the other regimes of interest – the unconditional bound on (currently at ), and on any further progress in asymptotic bounds for for larger – but the current focus is certainly on the bound on on GEH, as we seem to be tantalisingly close to an optimal result here.

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page (which has recently returned to full functionality, after a partial outage). In particular, the upper bound for has been shaved a little from to , and we have very recently achieved the bound on the generalised Elliott-Halberstam conjecture GEH, formulated as Conjecture 1 of this paper of Bombieri, Friedlander, and Iwaniec. We also have explicit bounds for for , both with and without the assumption of the Elliott-Halberstam conjecture, as well as slightly sharper asymptotics for the upper bound for as .

The basic strategy for bounding still follows the general paradigm first laid out by Goldston, Pintz, Yildirim: given an admissible -tuple , one needs to locate a non-negative sieve weight , supported on an interval for a large , such that the ratio

is asymptotically larger than as ; this will show that . Thus one wants to locate a sieve weight for which one has good lower bounds on the numerator and good upper bounds on the denominator.

One can modify this paradigm slightly, for instance by adding the additional term to the numerator, or by subtracting the term from the numerator (which allows one to reduce the bound to ); however, the numerical impact of these tweaks have proven to be negligible thus far.

Despite a number of experiments with other sieves, we are still relying primarily on the Selberg sieve

where is the divisor sum

with , is the level of distribution ( if relying on Bombieri-Vinogradov, if assuming Elliott-Halberstam, and (in principle) if using Polymath8a technology), and is a smooth, compactly supported function. Most of the progress has come by enlarging the class of cutoff functions one is permitted to use.

The baseline bounds for the numerator and denominator in (1) (as established for instance in this previous post) are as follows. If is supported on the simplex

whenever is supported on and is non-vanishing, then one can prove whenever

We can improve this baseline in a number of ways. Firstly, with regards to the denominator in (1), if one upgrades the Elliott-Halberstam hypothesis to the generalised Elliott-Halberstam hypothesis (currently known for , thanks to Motohashi, but conjectured for ), the asymptotic (2) holds under the more general hypothesis that is supported in a polytope , as long as obeys the inclusion

examples of polytopes obeying this constraint include the modified simplex

With regards to the numerator, the asymptotic (3) is valid whenever, for each , the marginals vanish outside of . This is automatic if is supported on , or on the slightly larger region , but is an additional constraint when is supported on one of the other polytopes mentioned above.

More recently, we have obtained a more flexible version of the above asymptotic: if the marginals vanish outside of for some , then the numerator of (1) has a lower bound of

Theorem 1 Suppose we can find and a function supported on a polytope obeying (4), not identically zero and with all marginals vanishing outside of , and with

Then implies .

In principle, this very flexible criterion for upper bounding should lead to better bounds than before, and in particular we have now established on GEH.

Another promising direction is to try to improve the analysis at medium (more specifically, in the regime ), which is where we are currently at without EH or GEH through numerical quadratic programming. Right now we are only using and using the baseline analysis, basically for two reasons:

We do not have good numerical formulae for integrating polynomials on any region more complicated than the simplex in medium dimension.

The estimates produced by Polymath8a involve a parameter, which introduces additional restrictions on the support of (conservatively, it restricts to where and ; it should be possible to be looser than this (as was done in Polymath8a) but this has not been fully explored yet). This then triggers the previous obstacle of having to integrate on something other than a simplex.

However, these look like solvable problems, and so I would expect that further unconditional improvement for should be possible.

This is the fourth thread for the Polymath8b project to obtain new bounds for the quantity

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on are:

(Maynard) Assuming the Elliott-Halberstam conjecture, .

(Polymath8b, tentative) . Assuming Elliott-Halberstam, .

(Polymath8b, tentative) . Assuming Elliott-Halberstam, .

(Polymath8b, tentative) . (Presumably a comparable bound also holds for on Elliott-Halberstam, but this has not been computed.)

(Polymath8b) for sufficiently large . Assuming Elliott-Halberstam, for sufficiently large .

While the bound on the Elliott-Halberstam conjecture has not improved since the start of the Polymath8b project, there is reason to hope that it will soon fall, hopefully to . This is because we have begun to exploit more fully the fact that when using “multidimensional Selberg-GPY” sieves of the form

with

where , it is not necessary for the smooth function to be supported on the simplex

but can in fact be allowed to range on larger sets. First of all, may instead be supported on the slightly larger polytope

However, it turns out that more is true: given a sufficiently general version of the Elliott-Halberstam conjecture at the given value of , one may work with functions supported on more general domains , so long as the sumset is contained in the non-convex region

and also provided that the restriction

is supported on the simplex

More precisely, if is a smooth function, not identically zero, with the above properties for some , and the ratio

is larger than , then the claim holds (assuming ), and in particular .

I’ll explain why one can do this below the fold. Taking this for granted, we can rewrite this criterion in terms of the mixed derivative , the upshot being that if one can find a smooth function supported on that obeys the vanishing marginal conditions

whenever and , and the ratio

is larger than , where

and

then holds. (To equate these two formulations, it is convenient to assume that is a downset, in the sense that whenever , the entire box lie in , but one can easily enlarge to be a downset without destroying the containment of in the non-convex region (1).) One initially requires to be smooth, but a limiting argument allows one to relax to bounded measurable . (To approximate a rough by a smooth while retaining the required moment conditions, one can first apply a slight dilation and translation so that the marginals of are supported on a slightly smaller version of the simplex , and then convolve by a smooth approximation to the identity to make smooth, while keeping the marginals supported on .)

We are now exploring various choices of to work with, including the prism

and the symmetric region

By suitably subdividing these regions into polytopes, and working with piecewise polynomial functions that are polynomial of a specified degree on each subpolytope, one can phrase the problem of optimising (4) as a quadratic program, which we have managed to work with for . Extending this program to , there is a decent chance that we will be able to obtain on EH.

We have also been able to numerically optimise quite accurately for medium values of (e.g. ), which has led to improved values of without EH. For large , we now also have the asymptotic with explicit error terms (details here) which have allowed us to slightly improve the numerology, and also to get explicit numerology for the first time.

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on are:

(Maynard) Assuming the Elliott-Halberstam conjecture, .

(Polymath8b, tentative) . Assuming Elliott-Halberstam, .

(Polymath8b, tentative) . Assuming Elliott-Halberstam, .

(Polymath8b) for sufficiently large . Assuming Elliott-Halberstam, for sufficiently large .

Much of the current focus of the Polymath8b project is on the quantity

where ranges over square-integrable functions on the simplex

with being the quadratic forms

and

It was shown by Maynard that one has whenever , where is the narrowest diameter of an admissible -tuple. As discussed in the previous post, we have slight improvements to this implication, but they are currently difficult to implement, due to the need to perform high-dimensional integration. The quantity does seem however to be close to the theoretical limit of what the Selberg sieve method can achieve for implications of this type (at the Bombieri-Vinogradov level of distribution, at least); it seems of interest to explore more general sieves, although we have not yet made much progress in this direction.

The best asymptotic bounds for we have are

which we prove below the fold. The upper bound holds for all ; the lower bound is only valid for sufficiently large , and gives the upper bound on Elliott-Halberstam.

For small , the upper bound is quite competitive, for instance it provides the upper bound in the best values

and

we have for and . The situation is a little less clear for medium values of , for instance we have

and so it is not yet clear whether (which would imply ). See this wiki page for some further upper and lower bounds on .

The best lower bounds are not obtained through the asymptotic analysis, but rather through quadratic programming (extending the original method of Maynard). This has given significant numerical improvements to our best bounds (in particular lowering the bound from to ), but we have not yet been able to combine this method with the other potential improvements (enlarging the simplex, using MPZ distributional estimates, and exploiting upper bounds on two-point correlations) due to the computational difficulty involved.

This is the second thread for the Polymath8b project to obtain new bounds for the quantity

either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on are:

(Maynard) .

(Polymath8b, tentative) .

(Polymath8b, tentative) for sufficiently large .

(Maynard) Assuming the Elliott-Halberstam conjecture, , , and .

Following the strategy of Maynard, the bounds on proceed by combining four ingredients:

Distribution estimates or for the primes (or related objects);

Bounds for the minimal diameter of an admissible -tuple;

Lower bounds for the optimal value to a certain variational problem;

Sieve-theoretic arguments to convert the previous three ingredients into a bound on .

Accordingly, the most natural routes to improve the bounds on are to improve one or more of the above four ingredients.

Ingredient 1 was studied intensively in Polymath8a. The following results are known or conjectured (see the Polymath8a paper for notation and proofs):

(Bombieri-Vinogradov) is true for all .

(Polymath8a) is true for .

(Polymath8a, tentative) is true for .

(Elliott-Halberstam conjecture) is true for all .

Ingredient 2 was also studied intensively in Polymath8a, and is more or less a solved problem for the values of of interest (with exact values of for , and quite good upper bounds for for , available at this page). So the main focus currently is on improving Ingredients 3 and 4.

For Ingredient 3, the basic variational problem is to understand the quantity

for bounded measurable functions, not identically zero, on the simplex

with being the quadratic forms

and

Equivalently, one has

where is the positive semi-definite bounded self-adjoint operator

so is the operator norm of . Another interpretation of is that the probability that a rook moving randomly in the unit cube stays in simplex for moves is asymptotically .

We now have a fairly good asymptotic understanding of , with the bounds

holding for sufficiently large . There is however still room to tighten the bounds on for small ; I’ll summarise some of the ideas discussed so far below the fold.

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